Density-aware Chamfer Distance as a Comprehensive Metric for Point Cloud Completion

TL;DR

This work identifies fundamental limitations of CD (insensitivity to local density) and EMD (global-structure emphasis, high cost) for evaluating and training on point clouds. It introduces Density-aware Chamfer Distance (DCD), a bounded, density-aware variant of CD that leverages per-point query frequency and an exponential distance term to better capture local structure while remaining efficient. DCD can serve as both an evaluation metric and a training objective, and is integrated into a two-stage point-cloud completion framework augmented by a learned point discriminator and guided down-sampling to balance density and preserve details. Experiments on MVP demonstrate that DCD offers more reliable, consistent correlations with visual quality than CD or EMD, and the proposed balanced design yields improvements across metrics with practical training benefits.

Abstract

Chamfer Distance (CD) and Earth Mover's Distance (EMD) are two broadly adopted metrics for measuring the similarity between two point sets. However, CD is usually insensitive to mismatched local density, and EMD is usually dominated by global distribution while overlooks the fidelity of detailed structures. Besides, their unbounded value range induces a heavy influence from the outliers. These defects prevent them from providing a consistent evaluation. To tackle these problems, we propose a new similarity measure named Density-aware Chamfer Distance (DCD). It is derived from CD and benefits from several desirable properties: 1) it can detect disparity of density distributions and is thus a more intensive measure of similarity compared to CD; 2) it is stricter with detailed structures and significantly more computationally efficient than EMD; 3) the bounded value range encourages a more stable and reasonable evaluation over the whole test set. We adopt DCD to evaluate the point cloud completion task, where experimental results show that DCD pays attention to both the overall structure and local geometric details and provides a more reliable evaluation even when CD and EMD contradict each other. We can also use DCD as the training loss, which outperforms the same model trained with CD loss on all three metrics. In addition, we propose a novel point discriminator module that estimates the priority for another guided down-sampling step, and it achieves noticeable improvements under DCD together with competitive results for both CD and EMD. We hope our work could pave the way for a more comprehensive and practical point cloud similarity evaluation. Our code will be available at: https://github.com/wutong16/Density_aware_Chamfer_Distance .

Figures (9)

• Figure 1: To generate the examples on the left, a complete yet noisy shape (blue) with $n$ points is combined with a partial yet clean shape (pink) with $2048$ points and then down-sampled back to 2048 points via FPS. Four point set distances are calculated between the generated shape and the ground truth. The results are averaged over the whole dataset to get the 3D histograms on the right: CD and HD are not sensitive to the mismatched density while highly influenced by noise when the intensity achieves a certain level; EMD and DCD share a similar pattern as $n$ changes, but DCD is more sensitive to the noise which also represents detailed structures.
• Figure 2: Comparison of assignment strategies and distance calculation. CD and DCD take the nearest neighbour locally, and DCD further considers the point-specific query frequency; EMD forces a one-to-one mapping, and the assigned pair of points may locate far from each other with weaker physical meaning.
• Figure 3: a presents the loss curves of CD-T, CD-P, and DCD and b presents the gradient curves of them; c and d visualizes the gradient for DCD with different $n$ and $\alpha$, respectively.
• Figure 4: a. visualization of mean shape over the dataset; b. visualization of $g(x)$ in Eqn. \ref{['eq:g_x']} at instance level; c. module structure of the point discriminator; d. two-stage framework with guided down-sampling.
• Figure 5: Comparison of CD, EMD, and DCD through examples and statistics. Below: examples from different methods that represent three typical cases: 1) EMD is low while CD is high; 2) CD is low while EMD is high; 3) both CD and EMD are low. DCD can only be lowered down in the third case. Upper left: A close shot at the cases above. Upper right: visualization of the three metrics on the test set, we can observe that 1) the positive correlation between CD and EMD is weak and the scatter points form a fan-shaped area, and 2) DCD (denoted by the point color) becomes lower towards the original point where both CD and EMD are low.